Arbitrage Free Value

Bonds can be valued either using the traditional valuation approach or the arbitrage-free valuation approach.

Under the traditional valuation approach, a single interest rate is used to discount all of a bond’s cash flows. In this approach, all cash flows of a bond are considered the same, regardless of their timing. In other words, each cash flow is viewed as just a token from the same package, and therefore the same discount rate is applied to all cash flows. This is a fundamental flaw because each individual cash flow of a bond is always unique. The use of a single discount rate in valuation may result in mispricing, thereby creating arbitrage opportunities. 

Under the arbitrage-free valuation approach, each cash flow is discounted at its own discount rate that takes into account the shape of the yield curve and the timing of the cash flow.

Perhaps an example will help illustrate. Consider a three-year U.S. Treasury note with a 10% coupon rate, paid semiannually. Considering the cash flows per $100 of par value, we would have six payments of $3 (one payment after every six months) and a final principal payment of $100.

• The traditional valuation approach would discount each of these cash flows using the same discount rate without considering their timing nor the shape of the yield curve. 
• The arbitrage-free valuation approach would view each cash flow as a zero-coupon instrument maturing on the date the cash flow is received. The three-year Treasury note would be viewed as a package of seven zero-coupon instruments maturing over a three-year period at six-month intervals. This would deny market participants an opportunity to realize an arbitrage profit by “stripping” the bond and selling the individual cash flows at a higher aggregate value than it would cost to purchase the Treasury in the market.

Arbitrage-Free Valuation of an Option-Free, Fixed-Rate Coupon Bond

An arbitrage-free value is the present value of expected future values using Treasury spot rates for option-free bonds. Arbitrage-free valuation usually involves three main steps:

• Step 1: Estimate the future cash flows.
• Step 2: Determine the appropriate discount rates that should
  be used to discount each cash flow.
• Step 3: Calculate the present value of the expected future cash flows by applying the appropriate discount rates determined in Step 2.

Thus, the following formula is used:

$$ PV=\frac{PMT}{\left(1+S_1\right)^1}+\frac{PMT}{\left(1+S_2\right)^2}+\ldots+\frac{PMT+FV}{\left(1+S_n\right)^N} $$

Where:

\(PMT\) is the periodic coupon.

\(FV\) is the face value.

\(S_1\), \(S_2,\) and \(S_N\) are the spot rates for periods 1 to \(N\).

Example: Arbitrage-Free Valuation

An 8% semi-annual coupon bond is priced at $800. It has a remaining term to maturity of 2 years. Given the following benchmark spot rates, the value of the bond, if its face value is $1,000, is closest to:

$$ \begin{array}{c|c} \textbf{Year} & \textbf{Spot Rates} \\ \hline 0.5 & 16\% \\ \hline 1 & 17\% \\ \hline 1.5 & 16\% \\ \hline 2 & 15\% \end{array} $$

Solution

$$ \text{Present value} =\frac{40}{1.08}+\frac{40}{{1.085}^2}+\frac{40}{{1.08}^3}+\frac{1,040}{{1.075}^4}=$881.52 $$

Note that $40 is the semi-annual coupon.

Question

An option free, 3-year 6% annual coupon bond priced at $100 has similar liquidity and risk to a Treasury bond whose par curve is shown in the table below.

$$ \textbf{Treasury Par Curve} \\ \begin{array}{c|c} \textbf{Term to Maturity (Years)} & \textbf{Par Rate} \\ \hline 1 & 3.00\% \\ \hline 2 & 4.00\% \\ \hline 3 & 5.00\% \end{array} $$

The arbitrage-free price for the bond is closest to:

1. $102.69.
2. $102.76.
3. $102.94.

Solution

The correct answer is B.

We first calculate the implied one-year spot rates given the above term structure by bootstrapping.

The one-year implied spot rate is 3%, as it is simply the one-year par yield.

We can bootstrap the two-year implied spot rate, \(r(2)\), as follows:

$$ \begin{align*} 1 &=\frac{0.04}{1.03}+\frac{(1+0.04)}{\left(1+r\left(2\right)\right)^2} \\ r\left(2\right)&=4.02\% \end{align*} $$

Similarly, the three-year spot rate can be bootstrapped by solving the equation:

$$ \begin{align*} 1 &=\frac{0.05}{1.03}+\frac{0.05}{{1.0402}^2}+\frac{1+0.05}{\left(1+r\left(3\right)\right)^3} \\ r\left(3\right) &=5.07\%  \end{align*} $$

The implied spot rate curve:

$$ \begin{array}{c|c|c} \textbf{Term to Maturity} & \textbf{Par Rate} & \textbf{Spot Rate} \\ \hline 1 & 3.00\% & 3.00\% \\ \hline 2 & 4.00\% & 4.02\% \\ \hline 3 & 5.00\% & 5.07\% \end{array} $$

To calculate the arbitrage-free price, each cash flow is discounted using the same maturity spot rate as the date of the cash flow.

$$ \text{Arbitrage-free price } (P)=\frac{6}{1.03}+\frac{6}{{1.0402}^2}+\frac{106}{{1.057}^3}=$102.76 $$

Reading 29: The Arbitrage-Free Valuation Framework

LOS 29(b) Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond.